## Defining parameters

 Level: $$N$$ = $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$10$$ Newform subspaces: $$27$$ Sturm bound: $$16384$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(384))$$.

Total New Old
Modular forms 4416 1776 2640
Cusp forms 3777 1680 2097
Eisenstein series 639 96 543

## Trace form

 $$1680 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} + O(q^{10})$$ $$1680 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} - 32 q^{10} - 16 q^{12} - 32 q^{13} - 8 q^{15} - 32 q^{16} - 16 q^{18} - 24 q^{19} - 4 q^{21} - 32 q^{22} + 16 q^{23} - 16 q^{24} - 8 q^{25} + 12 q^{27} - 32 q^{28} + 32 q^{29} - 16 q^{30} + 16 q^{31} - 32 q^{34} + 48 q^{35} - 16 q^{36} + 12 q^{39} - 32 q^{40} + 32 q^{41} - 16 q^{42} - 8 q^{43} + 4 q^{45} - 32 q^{46} - 16 q^{48} - 104 q^{49} - 96 q^{50} - 16 q^{51} - 224 q^{52} - 64 q^{53} - 48 q^{54} - 88 q^{55} - 224 q^{56} - 84 q^{57} - 320 q^{58} - 64 q^{59} - 208 q^{60} - 160 q^{61} - 192 q^{62} - 48 q^{63} - 416 q^{64} - 128 q^{65} - 208 q^{66} - 104 q^{67} - 192 q^{68} - 92 q^{69} - 416 q^{70} - 64 q^{71} - 16 q^{72} - 168 q^{73} - 224 q^{74} - 64 q^{75} - 288 q^{76} - 64 q^{77} - 112 q^{78} - 48 q^{79} - 96 q^{80} - 56 q^{81} - 32 q^{82} - 16 q^{84} - 112 q^{85} - 68 q^{87} - 32 q^{88} - 16 q^{90} - 24 q^{91} - 64 q^{93} - 32 q^{94} - 16 q^{96} - 64 q^{97} - 76 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(384))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
384.2.a $$\chi_{384}(1, \cdot)$$ 384.2.a.a 1 1
384.2.a.b 1
384.2.a.c 1
384.2.a.d 1
384.2.a.e 1
384.2.a.f 1
384.2.a.g 1
384.2.a.h 1
384.2.c $$\chi_{384}(383, \cdot)$$ 384.2.c.a 4 1
384.2.c.b 4
384.2.c.c 4
384.2.c.d 4
384.2.d $$\chi_{384}(193, \cdot)$$ 384.2.d.a 2 1
384.2.d.b 2
384.2.d.c 4
384.2.f $$\chi_{384}(191, \cdot)$$ 384.2.f.a 4 1
384.2.f.b 4
384.2.f.c 4
384.2.f.d 4
384.2.j $$\chi_{384}(97, \cdot)$$ 384.2.j.a 8 2
384.2.j.b 8
384.2.k $$\chi_{384}(95, \cdot)$$ 384.2.k.a 12 2
384.2.k.b 12
384.2.n $$\chi_{384}(49, \cdot)$$ 384.2.n.a 32 4
384.2.o $$\chi_{384}(47, \cdot)$$ 384.2.o.a 56 4
384.2.r $$\chi_{384}(25, \cdot)$$ None 0 8
384.2.s $$\chi_{384}(23, \cdot)$$ None 0 8
384.2.v $$\chi_{384}(13, \cdot)$$ 384.2.v.a 512 16
384.2.w $$\chi_{384}(11, \cdot)$$ 384.2.w.a 992 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(384))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(384)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(128))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(192))$$$$^{\oplus 2}$$